International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 8392148, 8 pages

http://dx.doi.org/10.1155/2016/8392148

## Research on the Effectiveness of Different Estimation Algorithm on the Autonomous Orbit Determination of Lagrangian Navigation Constellation

^{1}College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing, China^{2}College of Astronomy and Space Science, Nanjing University, Nanjing, China^{3}Beijing Satellite Navigation Center, Beijing, China

Received 7 July 2016; Revised 5 September 2016; Accepted 25 September 2016

Academic Editor: Enrico C. Lorenzini

Copyright © 2016 Youtao Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The accuracy of autonomous orbit determination of Lagrangian navigation constellation will affect the navigation accuracy for the deep space probes. Because of the special dynamical characteristics of Lagrangian navigation satellite, the error caused by different estimation algorithm will cause totally different autonomous orbit determination accuracy. We apply the extended Kalman filter and the fading–memory filter to determinate the orbits of Lagrangian navigation satellites. The autonomous orbit determination errors are compared. The accuracy of autonomous orbit determination using fading-memory filter can improve 50% compared to the autonomous orbit determination accuracy using extended Kalman filter. We proposed an integrated Kalman fading filter to smooth the process of autonomous orbit determination and improve the accuracy of autonomous orbit determination. The square root extended Kalman filter is introduced to deal with the case of inaccurate initial error variance matrix. The simulations proved that the estimation method can affect the accuracy of autonomous orbit determination greatly.

#### 1. Introduction

Deep space exploration has become a hot spot of aerospace. Several deep space probes have been launched. The autonomous navigation is important for deep space probes to deal with communication delay as well as reducing the dependency on ground stations. As early as 1968, the sextant had been used for autonomous navigation in “Apollo program” [1]. In 1999, “Deep Space 1” achieved autonomous orbit determination by tracking small celestial bodies with an optical sensor, which is the first successful on-orbit application of the deep space autonomous navigation technology [2]. The comet probe “Deep Impact” which was launched in 2005 also carried out its navigation and control automatically based on an optical navigation system with a high resolution imager [3]. In [4], Downs proposed to use X-ray pulsar radiation signal to navigate spacecraft. The rotation period of X-ray pulsar is extremely stable; therefore, time and the location of spacecraft can be determined by tracking several X-ray pulsars with given and fixed frequency [5]. The satellite navigation constellation can also provide navigation information for deep space probes. GPS can navigate the deep space probes when they are running in low-Earth orbits and medium Earth orbits. For deep space transfer orbits and deep space target orbits, the GPS is not good enough. Several researchers investigated weak GNSS signal navigation for the deep space probes [6–8]. Witternigg et al. introduced how GPS and Galileo could be used for orbit determination in future missions to the Moon [8]. Farquhar introduced a concept of using Earth-Moon libration point satellites for lunar navigation [9, 10]. In 2005, Hill suggested placing navigation constellation on the periodic orbits in the vicinity of libration points of the Earth-Moon system to support deep space navigation [11]. Zhang and Xu analyzed the architecture and navigation performance of the Lagrangian point satellite navigation system [12–14]. The Lagrangian navigation constellation is introduced to navigate the deep space probes autonomously. Hence, the navigation constellation itself should have the ability of autonomous orbit determination (AOD). The methods introduced in [1–5] can be considered as absolute navigation (or absolute autonomous orbit determination) because the estimated orbit refers to an inertial or quasi-inertial frame. Methods introduced in [6–14] can be classified as relative navigation. Relative navigation seeks optimal estimates for the position and velocity of one satellite relative to the other one.

Relative navigation usually is applied to satellites in a formation or constellation involved using GPS which restricts the spacecraft formation to near-Earth applications, such as Deep Space Mission 3 [15] and Grace project [16]. Relative navigation is primarily proposed for formation configuration control and formation reconfiguration. However, Lagrangian navigation constellation should provide absolute navigation information to deep space probes to achieve absolute navigation. Therefore, autonomous orbit determination of the Lagrangian navigation constellation is actually using the relative measurement to achieve absolute navigation. The laser interferometer space antenna (LISA) mission is an example which uses the relative range to assist the absolute orbit determination. The LISA mission which consists of three spacecraft separated by 5 million kilometers forming an equilateral triangle is a huge Michelson interferometer in space for gravitational wave detection [17]. The deep space-network provides a raw estimation of the absolute positioning for the three satellites. An accurate relative positioning will be provided with a laser-based ranging measurement in order to obtain an accuracy of positioning of tens of meters [18, 19]. Psiaki [20] and Markley [21] suggested using crosslink range, attitude information, and an optical tracker to determine the orbits autonomously. Yim et al. proposed using optical tracking and attitude information to find the direction vector between the two spacecraft and determine both orbits [22]. But these methods require extensive hardware development. In order to reduce the operational cost, size, and weight of spacecraft for formation missions, crosslink range can be used as the only measurement for orbit determination of a constellation. However, for the Earth navigation satellite constellation, there is a rank deficiency problem when only crosslink range is used to determine the orbit [23, 24]. Hill’s study illustrated that the rank deficiency problem does not exist for the Lagrangian navigation satellites because of the special dynamics near the libration points [11]. Thus, the Lagrangian navigation satellites can autonomously determine their orbits using only crosslink range. In [11], Hill discussed the unique distribution of Lagrangian orbit from the view of dynamics, which theoretically proved the autonomy of the Lagrangian navigation constellation. From the perspective of identifiability of epoch state, Qian et al. verified the feasibility of AOD for satellites in quasiperiodic orbits about the Earth-Moon libration point [25]. Based on circular restricted three-body problem (CR3BP), Du et al. researched the autonomous orbit determination method of satellites in halo orbits, and only crosslink range was used as observation [26]. In [23], Gao et al. discussed the feasibility of autonomous orbit determination using only the crosslink range measurement for a combined Lagrangian navigation constellation and GNSS. The most widely used algorithm for Lagrangian navigation satellite is extended Kalman filter (EKF) method. However, EKF is based on linearization of the system dynamics and the assumption of Gaussian process/measurement noise. These can seriously affect the performance of the state estimation and even lead to divergence. Unscented Kalman filter (UKF) which is based on the unscented transform can achieve higher accuracy than EKF while the added computational cost is not significant. More importantly, UKF is robust with respect to the initial conditions. Therefore, Sun et al. introduced UKF to relative navigation for multiple spacecraft formation flying [27]. Giannitrapani et al. analyze the performance of EKF and UKF for the localization of a spacecraft [28]. In order to improve the robustness and stability accuracy, Wang and Gu applied fault tolerant UKF in autonomous determination of relative orbit for satellite formation flying [29]. Rigatos introduced the technical analysis and implementation cost assessment of sigma-point Kalman filtering and particle filtering in autonomous navigation systems [30]. Reali and Palmerini provided a preliminary comparison of different estimation techniques to be used in formation flying navigation [31].

Since the CR3BP is sensitive to the state error and calculation error, the AOD of Lagrangian navigation satellite may refer to the accuracy of the estimation algorithms. One factor which must be considered in the AOD of Lagrangian navigation constellation is to prevent the divergence of AOD error. Therefore, we introduce four estimation methods to achieve the AOD of Lagrangian navigation constellation to analyze the effect on AOD by estimation method.

#### 2. Dynamical Model of Lagrangian Navigation Satellite

For satellites in Lagrangian point orbits, the equation of CR3BP should be an appropriate model to describe the satellites’ dynamical characteristics. Consider two massive bodies and moving under the action of just their mutual gravitation, and let their orbit around each other be a circle of radius . As shown in Figure 1, a noninertial, comoving frame of reference* o-xyz* is defined. The origin of frame* o-xyz* lies at the center of mass of the two-body system. The positive direction goes from to . The positive -axis is parallel to the velocity vector. The -axis is perpendicular to the orbital plane. Now the third body of mass which is vanishingly small compared to the primary masses and is introduced. We assume that the mass is so small that it has no effect on the motion of the primary bodies. This is called the restricted three-body problem.